Skipping Stones Gets Scientific

When children innocently ask head-scratching questions such as "Why is the sky blue?", most parents manage only some sheepish hems and haws. But when physicist Lydéric Bocquet's 7-year-old son asked him why a good throw makes a stone skip on a lake instead of sinking, Bocquet took the question as a challenge. He has now produced a set of equations explaining the physics underlying the popular pastime.

Two key forces act on a skipping stone: gravity, which pulls it down, and lift, the reactive force of the water, which pushes the stone up each time it hits the surface. If the lift force is greater than the force of gravity then the stone bounces up; otherwise it sinks. Bocquet's physical model bears out the conventional wisdom that the best skipping stones are flat and should be hurled nearly parallel to the water, fast and spinning. The stone's flatness maximizes the lift, as does its speed, which also provides energy to keep it bouncing along. These are the same factors that keep a water-skier from sinking, Bocquet says. Spin prevents the stone from tilting and hitting the water edge-first, just as fast rotations stabilize a bicycle or a spinning top.

According to the equations, a speed of 5 meters per second suffices to make a stone bounce six times. To bounce 38 times--the current world record--a stone must travel at about 12 meters per second. "That's actually not so fast," Bocquet says, but the thrower must also put on a spin of 15 revolutions per second, which is quite difficult, he says. "The small kick at the end of the throw is what's crucial if you want to set a world record," says Bocquet, a faculty member at Université Claude Bernard Lyon in France, who normally studies statistical mechanics. Students at the École Polytechnique in Paris are now building a catapult to test the numerical predictions of Bocquet's study, which will appear in the February issue of the American Journal of Physics.

"I think it's rather neat," says Bernie Nickel, a physicist at the University of Guelph in Canada. "The main points the author makes seem quite valid." But the model leaves out some complicating factors, cautions Jerry Gollub, a physicist at Haverford College in Pennsylvania--for instance, it ignores the effect of wind and the complex fluid flow around the stone. But the paper is interesting, he says. "It is always nice to see an everyday phenomenon analyzed quantitatively."